Problem: $ C = \left[\begin{array}{rr}4 & -1 \\ 5 & 0 \\ 0 & 2\end{array}\right]$ $ D = \left[\begin{array}{rr}3 & 5 \\ 2 & 1\end{array}\right]$ What is $ C D$ ?
Answer: Because $ C$ has dimensions $(3\times2)$ and $ D$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ C D = \left[\begin{array}{rr}{4} & {-1} \\ {5} & {0} \\ \color{gray}{0} & \color{gray}{2}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{5} \\ {2} & \color{#DF0030}{1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2} & ? \\ {5}\cdot{3}+{0}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2} & {4}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{1} \\ {5}\cdot{3}+{0}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2} & {4}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{1} \\ {5}\cdot{3}+{0}\cdot{2} & {5}\cdot\color{#DF0030}{5}+{0}\cdot\color{#DF0030}{1} \\ \color{gray}{0}\cdot{3}+\color{gray}{2}\cdot{2} & \color{gray}{0}\cdot\color{#DF0030}{5}+\color{gray}{2}\cdot\color{#DF0030}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}10 & 19 \\ 15 & 25 \\ 4 & 2\end{array}\right] $